Characterization of $\mathbb{Q}$ in $\mathbb{Q}_p$

Question

Let’s assume $\mathbb{Z}_p$ is the ring of p-adic integers. Then, how can we determine the elements of $\mathbb{Q}$ as an element of $\mathbb{Z}_p$, when it is defined? More specifically, I want to show that any p-adic integer with periodic p-adic expansion belongs to $\mathbb{Q}$, but it is not clear for me what I should show.

Answer 1

Let $$G = \{ \sum_{n\ge -N} a_n p^n \in \Bbb{Q}_p, a_n\in 0\ldots p-1,\exists m,k, \forall n\ge m, a_n= a_{n+k}\}$$

• Show it is a group using finiteness of states during carry calculations in $\sum_{n\ge -N} a_n p^n-\sum_{n\ge -N} b_n p^n$

• For $s=p^{-N} u/v\in \Bbb{Q}, p\nmid v$ let $r$ be the order of $p\bmod v$ then $$s=p^{-N} \frac{w}{1-p^r}= w p^{-N}\sum_{n\ge 0} p^{rn}\in wG\subset G$$

• Conversely if $\sum_{n\ge -N} a_n p^n\in G$ then $$\sum_{n\ge -N} a_n p^n=\sum_{n=-N}^{km-1} a_n p^n + \sum_{l=0}^{k-1} a_l \sum_{n=0}^\infty p^{l+kn+km}=\sum_{n=-N}^{km-1} a_n p^n + \sum_{l=0}^{k-1} a_l\frac{p^{l+km}}{1-p^k}\in \Bbb{Q}$$